The P3 Intersection Graph

We define a new graph operator called the P3 intersection graph, P3(G)- the intersection graph of all induced 3-paths in G. A characterization of graphs G for which P-3 (G) is bipartite is given . Forbidden subgraph characterization for P3 (G) having properties of being chordal , H-free, complete are also obtained . For integers a and b with a > 1 and b > a - 1, it is shown that there exists a graph G such that X(G) = a, X(P3( G)) = b, where X is the chromatic number of G. For the domination number -y(G), we construct graphs G such that -y(G) = a and -y (P3(G)) = b for any two positive numbers a > 1 and b. Similar construction for the independence number and radius, diameter relations are also discussed.

The Edge C4 Graph of a Graph

Abstract. The edge C4 graph E4(G) of a graph G has all the edges of Gas its vertices, two vertices in E4(G) are adjacent if their corresponding edges in G are either incident or are opposite edges of some C4. In this paper, characterizations for E4(G) being connected, complete, bipartite, tree etc are given. We have also proved that E4(G) has no forbidden subgraph characterization. Some dynamical behaviour such as convergence, mortality and touching number are also studied

Strongly distance-balanced graphs and graph products

A graph G is strongly distance-balanced if for every edge uv of G and every i 0 the number of vertices x with d.x; u/ D d.x; v/ 1 D i equals the number of vertices y with d.y; v/ D d.y; u/ 1 D i. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distance-balanced. It is also proved that connected components of the direct product of two bipartite graphs are strongly distancebalanced if and only if both factors are strongly distance-balanced. Additionally, a new characterization of distance-balanced graphs and an algorithm of time complexity O.mn/ for their recognition, wheremis the number of edges and n the number of vertices of the graph in question, are given

On The Center Sets and Center Numbers of Some Graph Classes

For a set S of vertices and the vertex v in a connected graph G, max x2S d(x, v) is called the S-eccentricity of v in G. The set of vertices with minimum S-eccentricity is called the S-center of G. Any set A of vertices of G such that A is an S-center for some set S of vertices of G is called a center set. We identify the center sets of certain classes of graphs namely, Block graphs, Km,n, Kn −e, wheel graphs, odd cycles and symmetric even graphs and enumerate them for many of these graph classes. We also introduce the concept of center number which is defined as the number of distinct center sets of a graph and determine the center number of some graph classes